Vectorial characterization of surface wave via one-dimensional photonic-atomic structure

Quantitative assessment of polarization properties of waves opens up the way for effective exploitation of them in many amazing applications. Tamm surface waves (TSW) that propagate on the interface of periodic dielectric media are proposed for many applications in numerous reports. The polarization state of TSW is not simply intuitive and would not be extracted from reflection spectra. Here considering orientation sensitive nature of the interaction between polarized electromagnetic wave and atom, we try to quantitatively characterize the polarization state of TSWs, excited on the surface of the 1D photonic crystal. To do this we performed direct contact between TSW and rubidium atomic gas by fabrication of a one-dimensional photonic crystal-atomic vapor cell and applied a moderate external magnetic field to create geometrical meaning and a sense of directionality to dark lines in reflection intensity. Our experimental results indicate that transition lines in the reflection spectrum of our hybrid system modify dependent on the orientation of the applied magnetic field and the transverse spin of TSW. We have used these changes to redefine the geometry of Voigt and Faraday for evanescent waves, especially Tamm surface waves. In the end, we performed simple mathematical operations on absorption spectra and extract the ratio of longitudinal and transverse electric field components of the polarization vector of TSW equal to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{2}{5}$$\end{document}25.

where the polarizable Polarizability tensor elements χ± and χ0 are related to susceptibility of atomic transitions with angular momentum ± 1 and 0(are associated with σ ± and π transitions.) The wave equation for an electromagnetic wave propagating in a non-magnetic dielectric medium based on Maxwell's equations can be derived.Here's the resulting wave equation: ε.  = 0 (eq s.3) assuming that the wavevector of propagative light makes an angle with the z-axis, direction of the external magnetic field, and with defining complex refractive index n as n 2 = ( c ω ) 2 . , we can write matrix form of wave equation as: for Faraday configuration, where ‖ , ( = ), by solving the eigenvalue problem for the dispersion matrix we obtained the refractive index and eigenvectors as and n 2 = √ε 0 (1 + χ − ) (eq s.6)In Voigt geometry, where the magnetic field axis is transverse to the light propagation axis ( ⊥ , ( = )), refractive index and eigenvectors obtain as: (eq s.9) and n 2 = √ε 0 (1 + χ 0 ) (eq s.10) ) and e 2 = ( 0 0 1 ) (eq s.12) The refractive indices (n 1 and n 2 ) and their associated eigenvectors correspond to different types of atomic transitions.n 1 corresponds to both σ+ and σ-transitions and n 2 corresponds to only π transitions.
e 1 represents elliptically polarized light in the plane perpendicular to ⃗B, and it equally drives both σ+ and σ-transitions.More clearly Let's assume that the incident light is linearly polarized along the y-axis in the laboratory frame.In the laboratory frame, this linear polarization can be described as a superposition of two circularly polarized components with equal amplitudes and opposite handedness, rotating in the X-Y plan.
Therefore, the linearly polarized light along the y-direction in the laboratory frame can be broken down into equal circular components in the atomic frame, which drive both σ+ and σ-transitions in Voigt geometry due to the orientation of the atomic quantization axis with respect to the external magnetic field.
e2 is associated with the field component that is parallel to the magnetic field (E z ⃦ B) and drives π transitions (Figure S1 b, c).As a general conclusion, if the electromagnetic field of incident light has a component perpendicular to the atomic quantization axis (magnetic field axis), it drives σ+ and σ-transitions.The field component that is aligned with the atomic quantization axis, drives the π transitions.As the Tamm surface wave has elliptical polarization in the xz plane, therefore, the component of the electric field of the incident light that is in the direction of the magnetic field drives π transitions.Also, the component of the electric field which is in the transverse of the magnetic field drives σ+ and σ-transitions (Figure S2).This analysis will provide a solid theoretical underpinning for the polarization state of the Tamm surface wave in our system [2].
Figure S3: schematic of our system Considering our system; the periodically nonhomogeneous photonic crystal which is in the direction normal to the interface, type A, occupies the half-space z > 0, and the anisotropic Rb vapor, type B, occupies the half-space z < 0, as shown in Figure S3.The constitutive relations are stated as z < 0 } (eq s.13) field phasors of p-polarized Tamm surface wave where e x (z), h y (z), e z (z) ≠ 0: z)x ̂+ e z (z)z ̂]exp(iqx) (eq s.15) () = h y (z)y ̂ exp(iqx) (eq s.16) Where q is complex wavenumber of TSW.By using Maxwell equations: For Z > 0, we insert eq s. 20 in eq s. 17 and eq s. 18 then obtain: For Z < 0, we insert eq s.21 in eq s. 17 and eq s. 18 then obtain: ωε zz B + iωμ 0 ) h y (z) (eq s. 24) The pair of eqs s. 22, 23 (as well as eqs s. 20, 21) yields the 2 × 2 matrix ordinary differential: where the column 2-vector (eq s. 27) and the matrix (eq s. 28) For Z < o and with considering dielectric tensor ε of hot vapor: (eq s. 30) , (eq s. 31) (eq s. 32) obeys the 2 × 2 matrix ordinary differential equation: We solve these equations (eq s. 33) near the interface of two media to demonstrate the elliptical polarization of Tamm surface waves.
For Z < o and using eq s. 21: For z˃0 and using eq s. 20 e x (z) = B 0 exp (−i√q 2 − ω 2 μ 0 ε A (z)z) (eq s. 37) e z (z) = iB 0 q √q 2 − ω 2 μ 0 ε A (z) exp (−i√q 2 − ω 2 μ 0 ε A (z)z) (eq s. 39) eqs s. 34, 36 show that in the TM polarization case, the electric field of TEW has two components that have a phase shift equal to π/2 with respect to each other.This results in elliptical polarization in the plane of incidence.

C. Effect of magnitude of external magnetic fields on spectral lines shift
In weak magnetic fields, the Zeeman splitting is relatively small, and the spectral lines associated with different sublevels may be overlapped.In strong magnetic fields, the Paschen-Back effect becomes significant.The energy levels of the atom can undergo substantial splitting, leading to distinct spectral lines.Frequency shift of these lines depends on magnitude of external magnetic fields [3].For example, the level hyperfine structure of the D1 line of 87Rb in the presence of a magnetic field is shown in Fig. 4S in the weak field (anomalous Zeeman) regime through the hyperfine Paschen-Back regime.In the Paschen-Back regime in both ground and excited states, one of the splitting of levels in F=2 has a redshift frequency compared to three other levels.This level contributes in transition of F=1, mF=-1(ground state;5 2 S 1/2 ) to F=2, mF=-2 (excited state; 5 2 P 1/2 ).∆m F =-1 so this transition is associated with σtransitions.This transition results in a distinct spectral line in the lower detuning frequency of the measured spectrum of the hybrid system.According to Fig. 2b (in manuscript), in the first Voight configuration where B is parallel to KB, the transverse electric field component of light drives σtransitions.In the second Voight configuration where B is parallel to , the longitude electric field component of light drives σtransitions.Therefore, each of the measured spectral lines in configurations provides us with unique information about the electric field components of the surface wave.
s.8) It is clear that σ ± transitions are excited by right/left circularly polarised light respectively.In the Faraday configuration, no polarized light cannot excite π transitions (Figure S1 a).

Figure
Figure S1: (a) Faraday and (b, c) Voight configuration for free space

Figure S2 :
Figure S2: Elliptical polarization of Tamm surface wave